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Proof step 29 from proof prepost-remove (previous step)

ar1[m] = elemindex(a, m2 + 1), kar[m2 + 1] = key-index(key, m), ar = ar1[kar[m1].idx, ar1[kar[m1].idx].idx:= (m2 + 1)], bh = bin-heap(kar[m2 + 1, kar[m1]], m1), # kar ≠ 1, #(keyset(kar, m1) ++ kar[m1]) ≠ 0, m1 ≠ m2 + 1, 2exp-size(# kar), m2 / 2 < # kar, kar[m1].key < kar[m2 / 2].key, m2 + 1 < # kar, m1 < # kar, kar[m1].idx < # ar1, m2 < m1, m2 < # kar, m2 + 1 < m1, 2 + m2 < # kar, heap-order(kar, m1 + 1), no-dups-idx(kar, m1 + 1), valid-idx(kar, ar1, m1 + 1), key-index(key, m) = kar[m1] ∨ key-index(key, m) ∈ keyset(kar, m1) ⊦ ⟪bin_min_heap_bubble_up#(m2 + 1; bh, ar)⟫ ( bh.size ≤ # bh.ar ∧ no-dups-idx(bh.ar, bh.size) ∧ valid-idx(bh.ar, ar, bh.size) ∧ 2exp-size(# bh.ar) ∧ heap-order(bh.ar, bh.size) ∧ id-idx(ar1, ar) ∧ keyset(bh.ar, bh.size) = (keyset(kar, m1) ++ kar[m1]) -- key-index(key, m))

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 30:
kar[m2 + 1] = key-index(key, m), ar1[m] = elemindex(a, m2 + 1), kar[m1] = key-index(key0, m0), ar = ar1[m0, ar1[m0].idx:= (m2 + 1)], bh = bin-heap(kar[m2 + 1, kar[m1]], m1), 2exp-size(# kar), 2 + m2 < # kar, m2 + 1 < m1, m2 < # kar, m2 < m1, m0 < # ar1, m1 < # kar, m2 + 1 < # kar, key0 < kar[m2 / 2].key, m2 / 2 < # kar, heap-order(kar, m1 + 1), no-dups-idx(kar, m1 + 1), valid-idx(kar, ar1, m1 + 1), key-index(key, m) = kar[m1] ∨ key-index(key, m) ∈ keyset(kar, m1) ⊦ ⟪bin_min_heap_bubble_up#(m2 + 1; bh, ar)⟫ ( bh.size ≤ # bh.ar ∧ no-dups-idx(bh.ar, bh.size) ∧ valid-idx(bh.ar, ar, bh.size) ∧ 2exp-size(# bh.ar) ∧ heap-order(bh.ar, bh.size) ∧ id-idx(ar1, ar) ∧ keyset(bh.ar, bh.size) = (keyset(kar, m1) ++ kar[m1]) -- key-index(key, m)), # kar = 1, #(keyset(kar, m1) ++ kar[m1]) = 0, m1 = m2 + 1

Rule argument

spec: key-index name: elim seq: ⊦ key = ki.key ∧ m = ki.idx ↔ ki = key-index(key, m) subst: [ki, key, m] → [kar[m1], key0, m0]

Additional information

Used simplifier rules: ⊦ key = ki.key ∧ m = ki.idx ↔ ki = key-index(key, m); (elim) Used terms: kar[m1], key0, m0